Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,12)(8,10)(15,19)(18,20)(21,22,26,27)(23,25,29,35)(24,31,32,28)(30,36,33,34);(5,6,9,11)(7,8,12,10)(14,17)(18,20)(22,27)(23,29)(24,32)(34,36);(5,7,9,12)(6,10,11,8)(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,33)(26,29)(27,35)(28,36)(30,32)(31,34);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(21,24,26,32)(22,28,27,31)(23,30,29,33)(25,34,35,36);(5,8)(6,7)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,29)(23,27)(24,34)(26,35)(28,33)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5186720);